陳逸聰91上期中
1.Why do electric circuits lead to differential equations?
  (5%)

2.(Half-life)If in a reactor, uranium 92U237 loses 10% of its weight within 1
  day, what is its half-life? How long would it take for 99% of the original
  amount to disappear?
  (5%)

3.Given the curves y=cx^2 , where c is arbitrary. Find their orthogonal
  trajectories.
  (10%)

4.(Mixing problem)The tank in Fig.1 contains 80 lb of salt dissolved in 500gal
      ╓─────────╖          of water. The inflow per minute is 20lb
  ══╝--                ║          of salt dissolved in 20 gal of water.
 ->       \               ║          The outflow is 20gal/min of the uniform
  ══╗    --------------║          mixture. Find the time when the salt
      ║                  ╚══      content y(t) in the tank reaches 95% of
      ║                       ->     its limiting value (as t→∞).
      ╰════════════
         Fig.1                        (10%)

5.Apply Picard's iteration to the following problem. Only do the first three
  steps. Sketch or plot every approximate solution curve obtained. Find the
  exact solution. Compare.      y'=y^2 ,y(0)=1
  (14%)

6.What is the superposition principle? Does it hold for nonlinear equations?
  For nonhomogeneous linear equations? For homogeneous linear equations? Why
  is it important?
  (3%)

7.How would you preatically test for linear independence of two functions? Of
  n solutions of a linear differential equation?
  (4%)

8.What is a particular solution? Why are particular solutions generally more
  common as final answers to practical problems than general solutions?
  (4%)

9.What is the Modification Rule and when did we need it?
  (3%)

10.Find a general solution for y'' - 2πy' + (π^2)y = 2e^(πx) .
   (Show the details of your calculations.)
   (8%)

11.Solve the problem. (D^3 - D^2 - D + 1)y=0,y(0)=2,y'(0)=1,y''(0)=0.
   (8%)

12.(a)Find the steady-state current in the RLC-circuit in Fig.2, assuming that
      L=1henry, R=2000ohms, C=4*10^-3 farad, and e(t)=110sin415t (66cycles/sec)
   (8%)      C                               ──
   ┌────||────┐                     k  spring
   │                  │                     │
   R                   L                      m  Mass
   │                  │                     │
   └──o E(t) o───┘                     c  Dashpot
           Fig.2                       Fig.3
   (b)Find a general solution of the homogeneous equation corresponding to the
      equation in (a).
   (8%)
   (c)Find the steady-state solution of the system in Fig.3 when m=1,c=2,k=6
      and the driving force is sin2t+2cos2t.
   (10%)

13.Solve the nonhomogeneous Euler-Cauchy equation by variation of parameters.
         (x^3)y''' - (3x^2)y'' + 6xy' -6y = (x^4)lnx
   (10%)

14.Find the currents I1(t) and I2(t) in the network shown in Fig.4, given the
    ┌─L=1henry─┬──||──┐ C=0.25farad     initial currents are
    │->I1     I1|│->I2 C I2|│                 I1(0)=28amperes,
 switch t=0      V│         V│                 I2(0)=14amperes.
    │            R1=4ohms    │
 E=12volts        │          │
    └──────┴─R2=6ohms┘   Fig.4
   (10%)

Total:120%
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